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In Griffiths, and Purcell and Morin, it is mentioned that the convention that the zero of potential is at infinity may fail when a charge distribution extends to infinity. In the problem of a point charge near a grounded conducting plane, when we don’t know how the charge induced on the conductor is distributed, how is it then certain that the potential approaches zero as the distance from the charge approaches infinity?

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You can answer this intuitively without calculating. Suppose the charge is positive. Then it will attract electrons from distant regions of the grounding plane.

There is a point in the plane nearest to the charge. Draw a circle around this point. The region inside will be negatively charged. Electrons will flow in until the repulsion of excess negative charge balances the attraction from the positive charge. At this point, electrons will stop flowing. The total force on any electron will be $0$.

Since the charge is above the plane, there is a component of force that is parallel to the plane. This attracts electrons. The perpendicular component does not. Also the distance to the charge is larger than the distance from an electron to nearby electrons. Both of these keep the density of electrons from becoming large. The total displaced charge inside any such circle is limited.

To a distant charge, force from the attracted electrons will be approximately parallel to the force from the charge. These will be equal and opposite. It will be as if there was a neutral ground plane plus a pair of equal and opposite charges.

The distant electron is sitting on a circle with a large radius. The total charge on the electrons that crossed the circle to create the excess charge inside adds up to that same as the positive charge.

The more distant a circle is, the larger its circumference. The smaller the number of electrons that cross an inch of circumference.

So any change to the distribution of electrons in the ground plane is biggest near the charge, and gets small far away. At infinite distances, there is no change.

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